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Hop41 In reply to markdow [2010-02-06 20:30:57 +0000 UTC]

Sorry it's taken me awhile to respond.

I think about circular chessboard's a lot so it's hard to make a brief comment. So I did a journal entry: [link]

This garden region tugs at me strongly. If the gravity I feel is any indication, this is quite a massive garden in which hides many wonders. Sadly I lack the time and energy to explore this place the way I want to.

Sometimes it makes me suffer vertigo. I become so disoriented that the things I think I observe are questionable.

This is also connected to the set of natural numbers as well as harmonic numbers 1/n. A set of points in a plane has translational symmetry if moving it by a vector (a,b) maps the set into itself. The first set I think of is a wallpaper based on square tiles. If you scale such a set by 1/n and add the smaller set to the original, the translational symmetry is preserved. For example this tile [link] seamlessly tiles the plane. I seem to recall you've done a similar perspective study.

Just as scaling a chessboard by 1/n doesn't destroy it's translational symmetry, the same is true of circular chessboards (I think). [link] Although I'm still not sure what translating (a, b) means on a circular chessboard.

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markdow In reply to Hop41 [2010-02-07 21:39:35 +0000 UTC]

Yes, I also feel the tug but only understand bits and pieces of the great puzzle. I've grown to enjoy the disorientation and just enjoy the ride, as a nice diversion when I can't concentrate on other things. I'd think that when I can concentrate I'd think about these things!

The "1/n" looping tile that I did is identical to yours, but only shows a single plane of points. There is a very satisfying way of understanding how sets of points line up in these figures (both in 2 and 3-D). Each allignment corresponds to a coprime pair of integers (don't share any common factor), and every member corresponds to an integer multiple of this coprime pair. I started to illustrate this (on this page is an attempt), but got distracted after starting a top down view. I really like simple geometric interpretations, but it's not easy showing them with clarity -- that's why I like your diagrams so much.

So I tend to think of this topic in terms of number theory (with a geometric interpretation), while you tend to think in terms of algebraic geometry , even though you might not know that's what your doing.

But we're definitely thinking about the same things. What you illustate here is a Diophantine equation . These problems are as old as old as geometry, but only recently (150 years) have math people realized that there is no general solution, but particular cases with geometric interpretations (like this) lead to interesting and deeper mathematical fields -- this is how algebraic geometry got started.

I think that translating (a,b) on a circular chessboard is not commutative: it depends on the order of the directions you translate. For example if the first coordinate refers to the radial direction and the second to circumferential, which way you go first matters. But I think you're right that scaling by 1/n doesn't alter the translational symmetry, as long as the order of translation is preserved.

In spherical geometry the "distances" (a,b)are in angular units, not Euclidean distance on a plane, and they are commutative. Going east then south is the same as going south then east iff the "distances" are in terms of latitude and longitude or similar orthogonal angular measures.

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